London-Paris Bachelier Workshop
September 19 –20, 2024, Amphi Hermite, IHP, Paris.

Thursday 19th September
2pm – 2:30pm Charles-Albert LEHALLE, Ecole polytechnique, Paris.
Title : Synthetic Data Generation For Portfolios Optimisation: Traps and counter measures.
Abstract : There are already a dozen of papers dedicated to generating synthetic data for portfolio optimization, ranging from risk-replicating methods for derivative pricing to those similar to Markowitz’s ones.
The goal of this talk is to identify the main mathematical reasons of the complexity of this generative task:
first the available sample is often small, and generating more data than what is available can provide a false
sentiment of confort, second I will show why the standard distance used by GANs is not adequate for portfolio construction that involves hedging of large variance components (that is often the case). I will make mybest to show where the memory should be located in the generate processes to provide meaningful results.
Last but not least, I will discuss that generating more than one time series can be needed, even to estimatea L-statistic (like a VaR or ES) on a one dimensional index. Finally, I will review the key features of theexisting papers and present preliminary results from a work with Rengim Cetingoz on a new generative model
designed to address many of these challenges.

2:30pm – 3pm Camilo GARCIA TRILLOS, University College London.
Title : A Conjugate Approach to solve a Stochastic Abatement Problem.
Abstract : We consider a stochastic pollution abatement model where we set ourselves to satisfy certain
environmental constraints with a high probability (e.g. satisfying the Paris agreement limit on global warming). The framework is reminiscent of the well-studied quantile hedging problem and can also be recast interms of a stochastic target model. However, we focus on the special case where only the drift is controlled.
Our main result shows that the solution can be obtained by optimizing over a family of associated conjugate
problems, which can be addressed using Backward Stochastic Differential Equations (BSDEs). Additionally,
we discuss how this connection can be leveraged to design effective numerical methods for solving our main
problem. This is a joint work with Yuxuan Wang.

13pm – 3:30pm Yating LIU, Paris Dauphine University.
Title : A statistical approach for simulating the density solution of a McKean-Vlasov
equation.
Abstract : We prove convergence results of the simulation of the density solution to the McKean-Vlasov
equation, when the measure variable is in the drift. Our method builds upon adaptive nonparametric resultsin statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel-type
estimator. In particular, we give a generalised Bernstein inequality for Euler schemes with interacting particles and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoreticalresults with a systematic numerical study and gather empirical evidence of the benefit of using high-orderkernels and data-driven smoothing parameters. This is a joint work with M. Hoffmann.

3:30pm – 4pm David ITKIN, London School of Economics and Political Science.
Title : Robust Maximization of Growth with Stochastic Factors and Price Stability.
Abstract : We consider a robust growth problem under model uncertainty and in the presence of both pricestability and (non-Markovian) stochastic covariance. Building on previous work of Kardaras & Robertson
we fix two inputs representing the instantaneous covariation and invariant density for the asset process, but
additionally allow these quantities to depend on the stochastic factor process. Our main result establishes
that, under mild technical conditions, the optimal strategy is given in feedback form by the gradient of afunction that solves a certain elliptic PDE. Remarkably, this function (and hence the optimal strategy) does
not depend on the stochastic factor. This provides a comprehensive answer to a question proposed by Fernholz in 2002 in the framework of Stochastic Portfolio Theory. We give two financial interpretations for thisresult and consider some examples. This talk is based on joint work with Benedikt Koch, Martin Larsson
and Josef Teichmann.

4pm - 4:30pm Break
4:30pm – 4.50pm Jiajie TAO, University College London.
Title : High Rank Path Development: an approach of learning the filtration of stochastic processes.
Abstract : Since the weak convergence for stochastic processes does not account for the growth of information over time which is represented by the underlying filtration, a slightly erroneous stochastic model in weak
topology may cause huge loss in multi-periods decision making problems. To address such discontinuities,
Aldous introduced the extended weak convergence, which can fully characterise all essential properties, including the filtration, of stochastic processes; however, it was considered to be hard to find efficient numerical
implementations. In this paper, we introduce a novel metric called High Rank PCF Distance (HRPCFD) for
extended weak convergence based on the high rank path development method from rough path theory, which
also defines the characteristic function for measure-valued processes. We then show that such HRPCFD
admits many favourable analytic properties which allows us to design an efficient algorithm for training
HRPCFD from data and construct the HRPCF-GAN by using HRPCFD as the discriminator for conditionaltime series generation. Our numerical experiments on both hypothesis testing and generative modelling
validate the out-performance of our approach compared with several state-of-the-art methods, highlighting2

its potential in broad applications of synthetic time series generation and in addressing classic financial andeconomic challenges, such as optimal stopping or utility maximisation problems.

4.50pm – 5:10pm Alicia BASSIERE, LGI, Centrale-Supélec, Université Paris-Saclay.
Title : A mean-field game of electricity market dynamics
Abstract : The global electricity sector’s environmental transition has led to significant growth in wind andsolar capacity and a decline in coal-based generation. However, this shift has also caused lower baseload electricity prices, impacting the profitability of conventional producers, and increased peakload price volatility,
making the system more reliant on natural gas. This paper proposes a long-term model using a mean-fieldgame (MFG) approach to describe the electricity industry’s dynamics, focusing on gas as a substitute for coal.
This paper extends the recent contribution [Aid, Dumitrescu, and Tankov, "The entry and exit game in theelectricity markets: A mean-field game approach," Journal of Dynamics & Games 8.4 (2021): 331] in several
ways, making the model much more realistic, especially for describing the medium-term impacts of energy
transition on electricity markets. We also present a central planner counterpart using mean-field control(MFC) and demonstrate their equivalence. A case study calibrated on German data examines the impact ofenvironmental policies. Our results show that without policies, gas power plants meet rising demand, leading
to higher gas and electricity prices. A carbon tax eliminates coal plants by 2030 and encourages renewables,
though it raises long-term electricity prices. Renewable subsidies promote wind power initially but are lesseffective beyond a certain threshold.

5:10pm – 5:30pm Yueying SUN, Department of Mathematics, London School of Economics andPolitical Science.
Title : Mandate Models and the Inelastic Market Hypothesis.
Abstract : Macroeconomic allocations in and out of the equity market lead to surprisingly large impacts on
stock valuations. Gabaix and Koijen study this phenomenon and provide a theoretical framework to explain
the observed price inelasticity. They assume that financial agents are constrained in their investment strategies and are subjected to a mandate that prescribes their investment allocations. Here we develop a rigorous
framework of a mandate model for a representative agent and provide precise conditions under which thestock valuation dynamics are well-defined. We also explore the aggregation of mandate functions from two
distinct mixed funds, each with distinct mandate function. The focus is on whether such aggregate mandate
function exists and can still yield a nice behaving equilibrium. While our analysis concerns two mixed funds,a successful aggregation here implies potential scalability to systems with multiple mixed funds. This is a
joint work with my supervisor Johannes Ruf.

5:30pm – 5:50pm Timothée FABRE, Laboratoire MICS, Centrale Supélec and SUN ZU Lab.
Title : Learning the Spoofability of Limit Order Books With Interpretable Probabilistic
Neural Networks.
Abstract : This paper investigates the detection of spoofing activity in small tick limit order books, focusing on cryptocurrency centralized exchanges. Inspired by the literature on Hawkes processes applied tomarket microstructure, we introduce novel order flow variables which incorporate not only the size of limit
orders but also their distance of placement in the order book. Using a market-by-order data set, we compute3

these features over observed limit orders, and we build a neural network model adapted to the predictionof the probability distribution of price movements. Our empirical analysis highlights the critical role thatthe distance of limit orders plays in the price formation process. Building on these insights, we develop
a spoofing detection framework where an agent uses a cost function to identify opportunities of profitable
market manipulation. Strikingly, our model indicates that spoofing the best quote is a poor decision. The
proposed detection algorithm is adapted to deployment in live trading environments, where it demonstrates
significant effectiveness in identifying suspicious trading activity. This work contributes to enhancing market
integrity by providing a robust tool for monitoring and mitigating spoofing in both cryptocurrency exchanges
and traditional financial markets.

5:50pm – 6:20pm Teemu PENNANEN, King’s College London.
Title : Dynamic programming and duality in convex stochastic optimization.
Abstract : We study a general class of convex stochastic optimization (CSO) problems that unifies many
common problem formulations from operations research, financial mathematics and stochastic optimal control. We extend the theory of dynamic programming and convex duality to allow for a unified and simplified
treatment of various special problem classes found in the literature. In particular, we establish the existenceof solutions to Bellman equations under quite general conditions that cover many instances of e.g. Markov decision processes where measurability questions are difficult in general. In addition, we establish the existenceof primal solutions and the absence of a duality gap without compactness or boundedness assumptions. In
the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage
condition and the reasonable asymptotic elasticity condition of the utility function. The existence of dual
solutions is established in the general format under conditions that extend those given by Rockafellar andWets for stochastic problems of Bolza. The extended theory allows also for significant generalizations toexisting problem formulations in financial mathematics and elsewhere. This is joint work with Ari-Pekka
Perkkio.

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Friday 20th September
9am – 9:30am Peter TANKOV, CREST, ENSAE-IP Paris.
Title : Optimal stopping and divestment timing under scenario ambiguity and learning.
Abstract : Aiming to analyze the impact of environmental transition on the value of assets and on asset
stranding, we study optimal stopping and divestment timing decisions for an economic agent whose future
revenues depend on the realization of a scenario from a given set of possible futures. Since the future scenario
is unknown and the probabilities of individual prospective scenarios are ambiguous, we adopt the smoothmodel of decision making under ambiguity aversion of Klibanoff et al (2005), framing the optimal divestment
decision as an optimal stopping problem with learning under ambiguity aversion. We then prove a minimax
result reducing this problem to a series of standard optimal stopping problems with learning. The theory is
illustrated with two examples: the problem of optimally selling a stock with ambigous drift, and the problem
of optimal divestment from a coal-fired power plant under transition scenario ambiguity.

9:30am – 10am Laura BALLOTA, Bayes Business School.
Title : Multivariate Additive Subordination with Applications in Finance.
Abstract : We introduce a tractable multivariate pure jump process in which the trading time is describedby an additive subordinator. The multivariate process retains the additivity property, and therefore is time
inhomogeneous, i.e., its increments are independent but nonstationary. We provide the theoretical frameworkof our process, perform a sensitivity analysis with respect to the time inhomogeneity parameters, and design
a Monte Carlo scheme to simulate the trajectories of the process. We then employ the model in the context of
option pricing in the FX market. We take advantage of the specific features of currency triangles to extract thejoint dynamics of FX log-rates. Extensive tests based on observed market data show that our model outperforms well established pure jump benchmarks. This is joint work with Giovanni Amici and Patrizia Semeraro.

10am – 10:30am Cyril BENEZET, LaMME, ENSIIE, UEVE UPS.
Title : Learning conditional distributions on continuous spaces.
Abstract : We investigate sample-based learning of conditional distributions on multi-dimensional unit
boxes, allowing for different dimensions of the input and output spaces. Our approach involves grouping datanear varying query points in the input space to create empirical measures in the output space. We employ two
distinct grouping methods: one based on a fixed-radius ball and the other on nearest neighbors. We establish
upper bounds for the convergence rates of both methods and, from these bounds, deduce optimal configurations for the radius and the number of neighbors. We propose to incorporate the nearest neighbor method
into neural network training, as our empirical analysis indicates an improvement in accuracy. For efficiency,
the training process utilizes approximate nearest neighbor search with random binary space partitioning.
Additionally, it employs the Sinkhorn algorithm and a sparsity-enforced transport plan. Our empirical findings demonstrate that, with a suitably designed structure, the neural network has the ability to adapt to asuitable level of Lipschitz continuity locally. This is a joint work with Ziteng Cheng and Sebastian Jaimungal.

10:30am – 11am Break5

11am – 11:20am Natascha HEY, Ecole Polytechnique & CFM.
Title : Trading with Concave Cross-Impact and Impact Decay.
Abstract : This paper investigates concave cross-impact models with impact decay, focusing on conditions
that rule out price manipulation. By formulating the profit and loss (PnL) in terms of alpha and Impact, we
solve a first instance of a concave cross-impact model. To prove applicability we present the first empirical
fit of an arbitrage free instance. Our results demonstrate that without manipulation constraints, the phasespace of models expands significantly. The empirical results confirm that cross-impact is, first, high whencorrelations are strong and, second, asymmetric when liquid assets impact illiquid ones.

11:20am – 11:40am Waleed TAOUM, King’s College London.
Title : Statistical modeling of SOFR term structure.
Abstract : SOFR derivatives market is still illiquid and incomplete so it is not amenable to classicalrisk-neutral term structure models which are based on the assumption of perfect liquidity and completeness.
We develop a statistical SOFR term structure model that is well-suited for risk management and derivatives
pricing within the incomplete markets paradigm. The model incorporates relevant macroeconomic factorsthat drive central bank policy rates which, in turn, cause random jumps often observed in the SOFR rates.
The model is easy to calibrate to historical data, current market quotes, and the user’s views concerning thefuture development of the relevant macroeconomic factors. The model is illustrated by indifference pricing
of SOFR derivatives. This is joint work with Teemu Pennanen.

11:40am – 12am Nathan De CARVALHO, LPSM, Université Paris Cité.
Title : Trading with propagators and constraints: applications to optimal execution
and battery storage.
Abstract : Motivated by optimal execution with stochastic signals, market impact and constraints in financial markets, and optimal storage management in commodity markets, we formulate and solve an optimal
trading problem with a general propagator model under linear functional inequality constraints. The optimal
control is given explicitly in terms of the corresponding Lagrange multipliers and their conditional expectations, as a solution to a linear stochastic Fredholm equation. We propose a stochastic version of the Uzawa
algorithm on the dual problem to construct the stochastic Lagrange multipliers numerically via a stochastic
projected gradient ascent, combined with a least-squares Monte Carlo regression step to approximate theirconditional expectations. We illustrate our findings on two different practical applications with stochastic
signals: (i) an optimal execution problem with transient impact, with either a ‘no-shorting’ constraint in thepresence of a ‘sell’ signal or a ‘no-buying’ constraint in the presence of a ‘buy’ signal; (ii) a battery storage
problem with seasonal signals and constraints on both the charging power and the load capacity of the battery.

12pm – 12:30pm Philippe BERGAULT, Université Paris Dauphine-PSL.
Title: Mean Field Games in a Stackelberg problem with an informed major player.
Abstract : We investigate a stochastic differential game in which a major player has a private information(the knowledge of a random variable), which she discloses through her control to a population of small players
playing in a Nash Mean Field Game equilibrium. The major player’s cost depends on the distribution of the
population, while the cost of the population depends on the random variable known by the major player. We

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show that the game has a relaxed solution and that the optimal control of the major player is approximatively
optimal in games with a large but finite number of small players. Joint work with Pierre Cardaliaguet andCatherine Rainer.

2pm – 2:30pm Laurence CARASSUS, MICS, Centrale-Supélec, Université Paris-Saclay.
Title : No free lunch for markets with multiple numeraires.
Abstract : We consider a new framework, that of a global market with a finite number of submarkets,
where there is a tradable numéraire for each submarket, but no tradable numéraire for the global market.
Under a global no arbitrage condition, we show the existence of a common density from which equivalent(local) martingale measures are constructed for each submarket. We also introduce several super-replication
prices, depending on the chosen type of hedging: on the global market, on a given submarket or on all submarkets separably. We prove duality results on these prices that allow to assess differences in characteristics
between the submarkets, such as liquidity or credit quality. The results are applied in concrete situations, inparticular in a Brownian setup.

2:30pm – 2:50pm Ruben WIEDEMANN, Imperial College London.
Title : Operator Deep Smoothing for Implied Volatility.
Abstract : We devise a novel method for implied volatility smoothing based on neural operators. The goalof implied volatility smoothing is to construct a smooth surface that links the collection of prices observed ata specific instant on a given option market. Such price data arises highly dynamically in ever-changing spatialconfigurations, which poses a major limitation to foundational machine learning approaches using classicalneural networks. While large models in language and image processing deliver breakthrough results on vast
corpora of raw data, in financial engineering the generalization from big historical datasets has been hinderedby the need for considerable data pre-processing. In particular, implied volatility smoothing has remained an
instance-by-instance, hands-on process both for neural network-based and traditional parametric strategies.
Our general operator deep smoothing approach, instead, directly maps observed data to smoothed surfaces.
We adapt the graph neural operator architecture to do so with high accuracy on ten years of raw intraday
S&P 500 options data, using a single set of weights. The trained operator adheres to critical no-arbitrage
constraints and is robust with respect to subsampling of inputs (occurring in practice in the context of outlier removal). We provide extensive historical benchmarks and showcase the generalization capability of ourapproach in a comparison with SVI, an industry standard parametrization for implied volatility.

2:50pm – 3:10pm Anna De CRESCENZO, LPSM, Université Paris Cité.
Title: Nonlinear Graphon mean-field systems.
Abstract : We address a system of weakly interacting particles where the heterogenous connections among
the particles are described by a graph sequence and the number of particles grows to infinity. Our results
extend the existing law of large numbers and propagation of chaos results to the case where the interaction
between one particle and its neighbors is expressed as a nonlinear function of the local empirical measure.
In the limit of the number of particles which tends to infinity, if the graph sequence converges to a graphon,then we show that the limit system is described by an infinite collection of processes and can be seen as aprocess in a suitable L2 space constructed via a Fubini extension. The proof is built on decoupling techniques

7and careful estimates of the Wasserstein distance.

3:10pm – 3:30pm Mingwei LIN, Department of Statistics, London School of Economics and Political Science.
Title : Equilibrium and market impact with uncertain insiders in limit order market.
Abstract : We consider the one-period Nash equilibrium among informed traders and competitive liquidity suppliers in the limit order market, where liquidity suppliers are uncertain about both the existence andquantity of informed traders in the market. We discuss how informed traders, if they exist, seek to exploit
their information advantage and how liquidity suppliers adjust the limit prices in equilibrium based on theirbeliefs about these informed traders. Our findings confirm the existence of the equilibrium, obtained througha fixed-point mapping problem when the explicit solution is unavailable for general distributions of asset
value and informed trader quantity. In equilibrium, we further show that fat-tailed trading asset returns leadto power law asymptotic market impact, while light tails cause logarithmic market impact. Furthermore, we
provide the numerically solvable fixed-point equations for the exponents and parameters in power law impact
and logarithmic impact.

3:30pm – 4pm Johannes MUHLE-KARBE, Imperial College London.
Title : A comparison of FX fixing methodologies.
Abstract : FX fixings are measurements of the exchange rate, published at regular intervals throughout
the day. They are widely used by market participants as benchmark rates. However, the methods by which
they are calculated differ between providers and have been changing over time. In this paper we provide aframework to assess competing fixing methodologies in terms of the dealer’s ability to hedge their fix exposureand the client’s effective transaction costs of executing at the fix. We demonstrate that the fixing window
width is the primary driver of dealer and client outcomes and that a weighting scheme by which prices areaveraged over the calculation window is only of secondary importance. A fixing window that is too narrowleads to excessive client transaction costs while a window that is too wide makes the hedging of fixing risk
commercially unviable for the dealer. This trade-off should be balanced by the fix administrators when they
settle on their preferred methodology. Joint work with Roel Oomen.
References:https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4857092
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4796356

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